Fractional Laplace operator in two dimensions,
approximating matrices, and related spectral analysis
Lidia Aceto1 · Mariarosa Mazza2 · Stefano Serra‑Capizzano3,4
Received: 24 October 2019 / Revised: 4 May 2020 / Accepted: 4 July 2020 © The Author(s) 2020
Abstract
In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by consid-ering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Fur-thermore, we give a concise description of the spectral properties when non-con-stant coefficients come into play. Several numerical experiments are reported and critically discussed.
Keywords Riesz fractional derivative operator · Toeplitz matrices · matrix transfer technique · spectral analysis · GLT theory
Mathematics Subject Classification 47A58 · 34A08 · 15B05 · 65F60 · 65F15 · 65F35
This work was funded by GNCS-INdAM. The authors are members of the INdAM research group GNCS.
* Lidia Aceto [email protected]
1 Department of Mathematics, University of Pisa, Via F. Buonarroti, 1/C, 56127 Pisa, Italy 2 Department of Science and High Technology, University of Insubria, via Valleggio 11,
22100 Como, Italy
3 Department of Humanities and Innovation, University of Insubria, via Valleggio 11,
22100 Como, Italy
4 Division of Scientific Computing, Department of Information Technology, Uppsala University,
1 Introduction
The study of the fractional Laplace operator was initiated in 1938 by Marcel Riesz
in his seminal work [37]. However, since then numerous definitions for such
opera-tor have been proposed due to the need of describing several distinct types of appli-cations (for a recent survey, refer to [24] and to the references therein). In the present paper, we review two main proposals to define the fractional Laplace operator which enable us to construct approximations in two or more spatial variables. As it is
well-known, the standard Laplacian in ℝd can be formulated as
For generalizing such operator towards the fractional case two different strategies can be followed:
(a) compute a fractional power of the d-dimensional Laplacian in (1);
(b) ‘fractionalize’ each integer order partial derivative in (1).
Obviously, these two strategies coincide in the 1D case, while they lead to differ-ent operators in more than one dimension. Formulation (a) is particularly useful when imposing boundary conditions which differ from the standard Dirichlet ones.
Indeed, in [21] Ilić et al. showed that the problems involving the fractional
Lapla-cian of the form (a) are well defined on finite domains with homogeneous bound-ary conditions of Dirichlet, Neumann and Robin type. This is due to the fact that the fractional Laplacian as in (a) has the same interpretation as the standard Lapla-cian in terms of spectral decomposition for such boundary conditions. Formulation (b) should be preferred instead when modeling phenomena that show a direction-dependent character and then require a different fractional order along each spatial dimension. In this regard, we mention the application of the fractional paradigm to
cardiac electrophysiology (see, e.g., [26]). Formulation (b) suits better than (a) here
because in the heart the myocyte fibers tend to align in a particular direction and therefore the propagation of the electrical wave is different orthogonally, longitudi-nally, and transversally along the fibers.
Discretizing (1) by means of finite differences with uniform step-size in each
direction and considering an hypercube as domain, we obtain a d-level Toeplitz
matrix associated to a generating function [10], also known as symbol [17, 18],
which provides an asymptotic description of the spectrum of the discretization matrices. In the light of formulations (a) and (b), for generalizing the symbol to the fractional case we can:
(1) take a fractional power of the symbol associated to the discretization of the
d-dimensional Laplacian in (1);
(2) ‘fractionalize’ the symbol associated to the discretization of each integer order partial derivative in (1). (1) 𝛥= d ∑ j=1 𝜕2 𝜕x2j.
Suitable (finite difference) approximations of the fractional Laplacian are then obtained by considering the corresponding d-level Toeplitz matrices (we refer the
reader to [30, 34, 35] for the unidimensional case). It is worth mentioning that a
similar idea was introduced by Lubich in [27, 28] for constructing the so-called
Fractional Linear Multistep Methods (FLMMs). In that case, the resulting schemes for a one-sided fractional operator (in the Riemann-Liouville or in the Caputo sense) were derived by considering a fractional power of the generating function of the coefficients of a classical linear multistep method for ordinary differential equations.
An alternative matrix representation of the fractional Laplacian can be obtained
by means of the so-called Matrix Transfer Technique (MTT) proposed in [21, 22].
The approximation to the fractional Laplacian in this case is given by a fractional power of the matrix representation obtained by using any reasonable method for
discretizing the standard Laplacian. As an application, we mention that in [45] the
MTT has been used for solving fractional Bloch-Torrey models in 2D with fractional Laplacian following both formulations (a) and (b). The interest in these models is due to the fact that they are used to study anomalous diffusion in the human brain. We point out that in order to face the high computational efforts that may result from this approach, rational approximations to the MTT and rational Krylov meth-ods have been recently introduced in [1–4, 8, 20].
The aim of this paper is twofold: first, we review the spectral properties of finite difference discretizations, which are of Toeplitz type associated with symbols, as in (i) and (ii); second, we investigate the spectral properties of the matrices obtained by the MTT as fractional powers of both finite difference and finite element approxima-tions of the standard Laplacian. Aside from the study of the spectrum/conditioning of the above matrices, we show that the corresponding matrix-sequences belong to the Generalized Locally Toeplitz (GLT) class.
One reason for characterizing all the aforementioned discretization matrix-sequences as GLT matrix-matrix-sequences relates to the solution of the corresponding lin-ear systems. Iterative schemes such as multigrid and preconditioned Krylov meth-ods—able to exploit the GLT structure—can be found in [13, 23, 25, 29, 32].
The rest of this paper is organized as follows. In Sect. 2 we briefly recall the main
results of the classical theory on the spectral behavior of Toeplitz matrices gener-ated by integrable functions, as well as the notion of symbol for a generic
matrix-sequence and the main properties of the GLT class. In Sect. 3 we collect various
definitions and results concerning the 1D fractional Laplacian. Section 4 is devoted
to the construction of matrix approximations to the fractional Laplacian both in one
and in two dimensions. Section 5 contains a detailed spectral analysis of the
underly-ing matrices. Few numerical tests that confirm our theoretical findunderly-ings are discussed
in Sect. 6. Finally, in Sect. 7 we draw some conclusions.
2 Preliminaries on Toeplitz and GLT matrix‑sequences
For an independent reading, in this section we formalize the definition of (d-level) Toeplitz matrix associated to a Lebesgue integrable function and we recall few key analytical features of the generating function, which provide information on the
spectral properties of the associated Toeplitz matrix (Sect. 2.1). Moreover, we intro-duce the GLT class, a ∗-algebra of matrix-sequences containing Toeplitz sequences, diagonal sampling matrix-sequences, and zero-distributed matrix-sequences (Sect. 2.2).
2.1 Spectral results for Toeplitz matrices
Let f be an integrable function defined on [−𝜋, 𝜋] and extended periodically to the whole real axis ℝ. We denote by
the Toeplitz matrix related to f (called generating funtion) whose coefficients aj,
j= 0, ±1, ±2, … , ±(n − 1), are the Fourier coefficients of f, i.e.,
In the case where f is an even, real-valued and nonnegative function, the related
Tn(f ) is a real, symmetric and positive semidefinite matrix. If f is only real-valued,
then the matrix Tn(f ) is Hermitian.
The following result allows to localize the spectrum of Tn(f ) in terms of ess inf f
and ess sup f , the essential infimum and the essential supremum of f (that is the
infi-mum and the supreinfi-mum of f, up to zero-measure sets); see, e.g., [17] for a proof.
Theorem 1 Let mf = ess inf f and Mf = ess sup f . If mf <Mf then the spectrum of Tn(f ) is contained in the interval (mf, Mf) independently of n ∈ ℕ . If mf = Mf then we have the trivial case where Tn(f ) = mfIn , with In denoting the identity of order n.
More generally, when f and g are real-valued and g is nonnegative and not iden-tically zero, the range of f/g gives precise indications regarding the spectrum of
Tn−1(g)Tn(f ).
Theorem 2 ([39, Theorem 2.1]) Let f, g be two real-valued integrable functions on
[−𝜋, 𝜋] such that
with r = ess inf(f ∕g) and R = ess sup(f ∕g). If g ≥ 0 (not identically zero) and r < R , then the spectrum of T−1
n (g)Tn(f ) is contained in the interval (r, R) independently of
n∈ ℕ . If r = R then we have the trivial case where T−1
n (g)Tn(f ) = rIn , with In denot-ing the identity of order n.
Tn(f ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ a0 a−1 … … a1−n a1 a0 a−1 ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ a1 a0 a−1 an−1 … … a1 a0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ ℂn×n aj= 1 2𝜋 ∫ 𝜋 −𝜋 f(𝜃) e−̂ij𝜃d𝜃, ̂i2= −1. 0 < r≤ f g ≤ R < ∞,
To provide a result concerning the behavior of the minimum eigenvalue of
Tn(f ) the following definitions must be introduced first.
Definition 1 Let f, g be two nonnegative integrable functions (not identically zero)
on 𝛺 ⊂ ℝd , d ≥ 1 . We write f ∼ g if and only if there exist two positive constants
0 < r≤ R < ∞ such that
uniformly on 𝛺.
Definition 2 Let f be a nonnegative integrable function on 𝛺 ⊂ ℝd , d ≥ 1 . We say
that f has a zero of order 𝛽 in 𝜃 = ̄𝜃 if and only if there exist two positive constants 0 < c≤ ̂c < ∞ such that
holds in a suitable neighborhood I ⊂ 𝛺 of ̄𝜃 . If I = 𝛺, then ̄𝜃 is also the unique zero of f and in this case the definition is equivalent to the relation f ∼ ‖𝜃 − ̄𝜃‖𝛽
2 ,
accord-ing to Definition 1.
Using the notations introduced in Definitions 1 and 2, the following result holds (see [9, 39]).
Theorem 3 Let f be a nonnegative integrable function defined on [−𝜋, 𝜋] such that ess inf f = 0 and ess sup f > 0. In addition, if for some ̄𝜃 ∈ [−𝜋, 𝜋] f satisfies the
con-dition f ∼ |𝜃 − ̄𝜃|𝛽, 𝛽 > 0 , then T
n(f ) is a positive definite matrix for any n ∈ ℕ and 𝜆min(Tn(f )) ∼ 1∕n𝛽.
We remark that a similar result holds true also for the maximum eigen-value; of course, in this case we should consider the minimum eigenvalue of
Tn(Mf − f ) = MfIn− Tn(f ) , according to the notations of Theorem 1.
Theorem 4 Let f, g be two nonnegative integrable functions on [−𝜋, 𝜋] such that ess inf f = 0, ess sup f > 0, ess inf g = 0, and ess sup g > 0. Assuming that
then,
The previous results can be extended to two-level Toeplitz matrices which are
defined as follows. Let f (𝜃1, 𝜃2) be an integrable function defined on the square
[−𝜋, 𝜋]2 and extended periodically on ℝ2. Denoting by
rg≤ f ≤ Rg c≤ f ‖𝜃 − ̄𝜃‖𝛽 2 ≤ ̂c 0 < r≤ f g ≤ R < ∞,
the Fourier coefficients of f, we define the two-level Toeplitz matrix Tn,m(f )
gener-ated by f as follows:
with
By the construction given in (2)–(4), we can easily deduce that when the
generat-ing function is a multiplicatively separable function, i.e., f (𝜃1, 𝜃2) = s1(𝜃1)s2(𝜃2) , we
have
with ⊗ being the standard Kronecker product [7]. The latter and the linearity of the
Toeplitz operator imply that for a generic integrable function f
where Jr,1 ( Jr,−1 ) denotes the lower (upper) shift matrix of order r.
The last representation suggests the following compact notation in the case of a d-variate generating function and of a d-level Toeplitz matrix. Let f (𝜽) ,
𝜽= (𝜃
1,… , 𝜃d) be an integrable function defined on [−𝜋, 𝜋]d and extended
peri-odically over ℝd . Denoting by
the Fourier coefficients of f, when d ≥ 2 we define the d-level Toeplitz matrix Tn(f )
generated by f as (2) aj,k= 1 4𝜋2∫ 𝜋 −𝜋∫ 𝜋 −𝜋 f(𝜃1, 𝜃2) e−̂i(j𝜃1+k𝜃2)d𝜃1d𝜃2, j, k∈ ℤ, (3) Tn,m(f ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ A 0 A−1 … … A1−n A 1 A0 A−1 ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ A 1 A0 A−1 An −1 … … A1 A0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (4) A j= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ aj,0 aj,−1 … … aj,1−m aj,1 ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ aj,−1 aj,m−1 … … aj,1 aj,0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (5) Tn,m(f ) = Tn(s1(𝜃1)) ⊗ Tm(s2(𝜃2)), Tn,m(f ) = ∑ (1−n,1−m)≤(j,k)≤(n−1,m−1) aj,kTn(êij𝜃1) ⊗ T m(êik𝜃2) = ∑ (1−n,1−m)≤(j,k)≤(n−1,m−1) aj,kJn|j| ,sign(j)⊗J |k| m,sign(k), aj= 1 (2𝜋)d ∫ [−𝜋,𝜋]d f(𝜽) e−̂i(j1𝜃1+…+jd𝜃d)d𝜽, j= (j 1,… , jd) ∈ ℤ d
where n = (n1,… , nd) , 1 = (1, … , 1) , and the ordering considered in 1 − n ≤ j ≤ n − 1 is the lexicographical one employed for j and with s ≤ t if and
only if sj≤ tj∀j = 1, … , d , s = (s1,… , sd), t = (t1,… , td) , see [42].
The generating function f provides a description of the spectrum of Tn(f ) , for n
large enough in the sense of the following definition.
Definition 3 Let f ∶ 𝛺 ⊂ ℝd → ℂ be a measurable function defined on a set 𝛺,
measurable in the sense of Lebesgue, with finite measure ̄𝜇(𝛺). Moreover, let us
assume that {An}n is a sequence of matrices with eigenvalues 𝜆j(An) and singular
values 𝜎j(An) , j = 1, … , dn, and that dn is the size of An with dn→ ∞ , as n → ∞ ,
i.e., ni→ ∞, i = 1, … , d.
• We say that {An}n is distributed as f over 𝛺 in the sense of the eigenvalues,
and we write {An}n∼𝜆(f , 𝛺), if
for every continuous function F with compact support. In this case, we say that f is the symbol of {An}n.
• We say that {An}n is distributed as f over 𝛺 in the sense of the singular values,
and we write {An}n∼𝜎(f , 𝛺), if
for every continuous function F with compact support. In particular, when f ≡ 0
we say that {An}n is a zero-distributed matrix-sequence.
In the following, for the sake of simplicity we will often write {An}n∼𝜆f,
{An}n∼𝜎f in place of {An}n∼𝜆(f , 𝛺), {An}n∼𝜎(f , 𝛺).
Remark 1 If f is smooth enough, an informal interpretation of the limit relation 6
(resp. 7) is that when n is sufficiently large, then the eigenvalues (resp. singular
val-ues) of An can be approximated by a sampling of f (resp. |f|) on a uniform equispaced
grid of the domain 𝛺.
For Toeplitz matrix-sequences, the following theorem (due to Szegö, Tilli,
Tyrtyshnikov, Zamarashkin, … ) holds, cf. [19, 41, 43].
Tn(f ) = ∑ 1−n≤j≤n−1 ajTn1(êij1𝜃1) ⊗ ⋯ ⊗ Tnd(êijd𝜃d) = ∑ 1−n≤j≤n−1 ajJ|j1| n1,sign(j1) ⊗ ⋯ ⊗J|jd| nd,sign(jd), (6) lim n→∞ 1 dn dn ∑ j=1 F(𝜆j(An)) = 1 ̄ 𝜇(𝛺) ∫𝛺 F(f (t))dt, (7) lim n→∞ 1 dn dn ∑ j=1 F(𝜎j(An)) = 1 ̄ 𝜇(𝛺) ∫𝛺 F(|f (t)|)dt,
Theorem 5 Let {Tn(f )}n be a Toeplitz sequence generated by f ∈ L1([−𝜋, 𝜋]d).
Then, {Tn(f )}n∼𝜎(f , [−𝜋, 𝜋]d). In addition, if f is real-valued, then {Tn(f )}n∼𝜆(f , [−𝜋, 𝜋]d).
2.2 GLT class
The formal definition of GLT matrix-sequences is rather technical and involves somewhat cumbersome notation. Therefore, in this subsection we briefly discuss some properties of the GLT class which are sufficient for our aims and we refer the
reader to [17] for more details. Throughout, we use the following notation
to say that the matrix-sequence {An}n is a GLT matrix-sequence with GLT symbol
𝜅(x, 𝜽) , where x = (x1,… , xd) are the physical variables and 𝜽 = (𝜃1,… , 𝜃d) are the
Fourier variables. First we introduce a certain type of diagonal matrices which, as we shall see, come into play when dealing with differential problems with variable coefficients.
Definition 4 Let 𝔞 ∶ [0, 1]d→ ℂ be a Riemann-integrable function. The diagonal sampling matrix of order n1⋯ nd is given by
where j∕n =(j1∕n1,… , jd∕nd
) .
We list below key features of GLT matrix-sequences which represent a character-ization of GLT matrix-sequences. In other words the ∗-algebra of matrix-sequences satisfying items GLT1–GLT6 coincides with the whole class of GLT matrix-sequences. We have chosen this way, not only for the sake of brevity and notational simplicity, but also because items GLT1–GLT6 furnish a constructive way for veri-fying whether a concrete matrix-sequence belongs to the GLT class.
GLT1 Let {An}n∼GLT𝜅. Then {An}n∼𝜎(𝜅, G) . If the matrices An are Hermitian,
then it also holds that {An}n∼𝜆(𝜅, G).
GLT2 The set of GLT matrix-sequences forms a ∗-algebra that is closed under
linear combinations, products, inversion, conjugation. In formulas, let {An}n∼GLT𝜅1 and {Bn}n∼GLT𝜅2 , then
• {𝛾1An+ 𝛾2Bn}n∼GLT𝛾1𝜅1+ 𝛾2𝜅2, 𝛾1, 𝛾2∈ ℂ;
• {AnBn}n∼GLT𝜅1𝜅2;
• {A−1
n }n∼GLT𝜅1−1 provided that 𝜅1 is invertible almost everywhere;
• {A∗
n}n∼GLT𝜅̄1.
GLT3 Any sequence {Tn(f )}n generated by a function f ∈ L1([−𝜋, 𝜋]d) is a GLT
matrix-sequence with symbol 𝜅(x, 𝜽) = f (𝜽).
{An}n∼GLT𝜅, 𝜅∶ G → ℂ, G= [0, 1]d× [−𝜋, 𝜋]d,
D
GLT4 Every diagonal sampling matrix-sequence {Dn(𝔞)}n is a GLT
matrix-sequence whose symbol is given by 𝜅(x, 𝜽) = 𝔞(x).
GLT5 Every zero-distributed matrix-sequence is a GLT matrix-sequence with
symbol 0 and viceversa, i.e., {An}n∼𝜎0 ⟺ {An}n∼GLT0.
GLT6 If {An}n∼GLT𝜅 and An is Hermitian for each n ∈ ℕd, then
{f (An)}n∼GLTf(𝜅) for every continuous function f ∶ ℂ → ℂ.
According to Definition 3, in the presence of a zero-distributed matrix-sequence the singular values of the n th matrix (weakly) cluster around 0. This is formalized in the following result.
Proposition 1 ([17, Theorem 3.2]) Let {An}n be a matrix-sequence with An of size
dn with dn→ ∞ , as n → ∞ . Then {An}n∼𝜎0 if and only if there exist two matrix-sequences {Rn}n and {En}n such that An= Rn+ En , and
3 The 1D fractional Laplacian
In this section we review two different definitions of the fractional Laplacian in one dimension and we report their basic properties useful in the sequel. We shall always assume that the fractional Laplacian is applied to a suitable smooth function, say v (for instance, v ∈ L1(ℝ) ). In addition, for simplicity, we denote the space variable by x.
Consider the Riesz potential operator that in the 1D case can be formulated as
where 𝛤 denotes the Gamma function. It verifies the following properties:
• lim𝛽→0+(𝕀𝛽v)(x) = v(x);
• 𝕀𝛽(𝕀𝛾v)(x) = (𝕀𝛽+𝛾v)(x), 0 < 𝛽, 𝛾, 𝛽+ 𝛾 < 1;
• −𝛥(𝕀𝛽+2v)(x) = (𝕀𝛽v)(x), 0 < 𝛽, 𝛽+ 2 < 1.
Such operator allows to introduce the first definition of fractional Laplacian to which we refer, also known as Riesz fractional derivative operator,
cf. [11, 38]. In [11, Lemma 2.2], under the hypotheses that v ∈ C5(ℝ) and all
deriva-tives up to order five belong to L1(ℝ), Çelik and Duman proved that
lim n→∞ rank(Rn) dn = 0, lim n→∞‖En‖2= 0. (𝕀𝛽v)(x) = 1 2 cos(𝛽𝜋∕2)𝛤 (𝛽) ∫ +∞ −∞ |x − y|𝛽−1v(y) dy, 𝛽 ∈ (0, 1), (8) −(−𝛥)𝛼∕2v(x) ∶= −(−𝛥)(𝕀2−𝛼v)(x), 𝛼∈ (1, 2], (9) −h−𝛼𝛥𝛼c 1v(x) = −(−𝛥)(𝕀 2−𝛼v)(x) + O(h2),
where 𝛥𝛼
c1 denotes a generalization of the integer-order centered differences to the
fractional case proposed by Ortigueira in [30, Definition 3.5], namely
Taking the limit as h approaches 0 of both sides of (9) gives another form for the
fractional Laplacian (see (8))
which in [30, Definition 4.1] was called type 1 fractional centred derivative.
There-fore, setting in (10)
we have
It can be easily verified that the coefficients 𝓁k satisfy the following properties [11,
Property 2.1]: 1. 𝓁k+1 = ( 1− 𝛼+1 𝛼∕2+k+1 ) 𝓁k;
2. 𝓁0 >0 and 𝓁−k= 𝓁k≤ 0 for each k ≥ 1.
From the relationship [33, (1.33) p. 8]
where z = 𝛽 + m, m < z < m + 1, 𝛽 ∈ (0, 1), for all nonnegative values of k we obtain
Consequently, for k ≥ 0 we can write (see [46])
or (10) 𝛥𝛼c 1v(x) = +∞ ∑ k=−∞ (−1)k𝛤(𝛼 + 1) 𝛤(𝛼∕2 − k + 1)𝛤 (𝛼∕2 + k + 1)v(x − kh), h∈ ℝ + . D𝛼c 1v(x) = limh→0h −𝛼𝛥𝛼 c1v(x), 𝓁k= (−1) k𝛤(𝛼 + 1) 𝛤(𝛼∕2 − k + 1)𝛤 (𝛼∕2 + k + 1), k= 0, ±1, ±2, … , (11) −(−𝛥)𝛼∕2v(x) = − lim h→0h −𝛼 +∞ ∑ k=−∞ 𝓁kv(x − kh). 𝛤(z)𝛤 (1 − z) = (−1) m𝜋 sin(𝜋𝛽) (−1)k 𝛤(𝛼∕2 − k + 1) = − sin(𝛼𝜋∕2) 𝜋 𝛤(−𝛼∕2 + k), −sin(𝛼𝜋∕2)𝛤 (𝛼 + 1) 𝜋 = 1 2 cos(𝛼𝜋∕2)𝛤 (−𝛼). 𝓁k= −𝛤(−𝛼∕2 + k)𝛤 (𝛼 + 1) 𝜋𝛤(𝛼∕2 + k + 1) sin(𝛼𝜋∕2),
By using the ratio expansion of two Gamma functions at infinity (see, e.g., [38, p. 17])
one deduces
For this reason, the series in (10) converges absolutely for a bounded function
v∈ L1(ℝ). By exploiting the decay of the coefficients 𝓁
k, a “short memory” version
of the scheme proposed by Ortigueira has been implemented in [36]. Finally, using
the Fourier definition of the fractional Laplacian, in [31] was proved that
In this case, the Fourier coefficients are given by
We conclude this section by pointing out that several applications require the solu-tion of fracsolu-tional differential equasolu-tions expressed in terms of fracsolu-tional Laplacian multiplied by a certain non-constant coefficient. In such cases, the spatial operator has the following form
with D a non-negative function. Later we also give spectral insights on this operator under the additional hypothesis that D is Riemann integrable.
4 Matrix forms for Laplace operators
Aiming to discretize the fractional Laplace operator in two (or more) dimensions,
in the following we always impose absorbing boundary conditions (see [12] for
fur-ther details). In the one-dimensional case this is equivalent to applying the fractional Laplace operator to functions of the form
Different boundary conditions can also be imposed and the spectral analysis in terms of distribution is quite simplified using the GLT machinery. In fact, the perturbation
𝓁k= 1 2 cos(𝛼𝜋∕2)𝛤 (−𝛼) 𝛤(−𝛼∕2 + k) 𝛤(𝛼∕2 + k + 1). 𝛤(z + r) 𝛤(z + s) = z r−s(1+ O(z−1)), 𝓁k∼|k|−𝛼−1, |k| ≫ 1. +∞ ∑ k=−∞
𝓁kêik𝜃 = (1 − êi𝜃)𝛼∕2(1 − e−̂i𝜃)𝛼∕2= (2 − 2 cos 𝜃)𝛼∕2.
(12) 𝓁k= 1 2𝜋 ∫ 𝜋 −𝜋 (2 − 2 cos 𝜃)𝛼∕2e−̂ik𝜃d𝜃. (13) −D(x)(−𝛥)𝛼∕2 w(x) = { v(x), x ∈ (a, b), 0, otherwise.
induced by a different choice of boundary conditions can be represented as a zero-distributed matrix-sequence and hence GLT5 combined with items GLT1-GLT2 represents the key for the related spectral analysis.
We finally emphasize that from a matrix viewpoint the method in Sect. 4.1
amounts in taking the Toeplitz structure generated by a proper ‘fractionalization’
of the symbol of the discrete Laplacian, while the method in Sect. 4.2 amounts in
taking the same fractional power applied to a (possibly Toeplitz) structure. It is evident that the first method is computationally much more convenient: the inter-esting fact is that the two resulting GLT sequences possess the same symbol (so that they differ for a zero-distributed matrix-sequence), even if they look quite different in terms of entries and of global algebraic structure (for instance in the second form the Toeplitz structure is completely lost).
4.1 Toeplitz matrix form
Consider the uniform mesh of the domain [a, b] with step-size h = (b − a)∕(n + 1), i.e.,
From (11), when 𝛼 = 2, it is clear that the standard 1D Laplacian is expressed in
terms of centered differences as
where we abbreviate wj= w(𝜉j). Setting
all these formulas can be written simultaneously in matrix form as −h−2T
n(f2).
We remark that Tn(f2) is the Toeplitz matrix whose entries on the kth diagonal
are the Fourier coefficients of the generating function
Raising f2 to the appropriate power yields
Therefore, from (12) we deduce that 𝓁k are the Fourier coefficients of f𝛼. Denoting
by (14) 𝜉j= a + jh, j= 0, 1, … , n + 1. d2w(x) dx2 || || |x=𝜉j = lim h→0 wj−1− 2wj+ wj+1 h2 , j= 1, 2, … , n, (15) Tn(f2) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 2 −1 −1 2 −1 ⋱ ⋱ ⋱ −1 2 −1 −1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ ℝn×n, (16) f2(𝜃) = 2 − 2 cos 𝜃. f𝛼(𝜃) =(f2(𝜃))𝛼∕2.
the Toeplitz matrix related to f𝛼, and taking into account (11) it is a simple matter
to verify that −h−𝛼T
n(f𝛼) represents the matrix form for the 1D fractional Laplacian
described in Sect. 3, cf. [11, 46].
Now we turn to the Laplace operator in two dimensions. Discretizing in space with a uniform mesh of step-size h in each domain direction and using the 5-point finite-difference stencil leads to the block tridiagonal matrix having the following form:
with In denoting the identity matrix of size n and Tn(f2) defined according to (15).
Tn,n(𝜑2) is a 2-level Toeplitz matrix which corresponds to the bivariate symbol (see
(2))
Bearing in mind the idea presented above for approximating the 1D fractional Lapla-cian, in the two-dimensional case we may approximate the fractional Laplacian by considering the matrix Tn,n(𝜑𝛼) related to the generating function
From (16), we can immediately rewrite 𝜑2(𝜃1, 𝜃2) = f2(𝜃1) + f2(𝜃2). Consequently,
Another way for constructing an approximation of the 2D fractional Laplacian is based on the 2-level Toeplitz matrix generated by the separable function
hereafter we denote this matrix by Tn,n(𝜓𝛼). Using (5), we obtain that
where Tn(f𝛼) is given by (17). Obviously, Tn,n(𝜓2) = Tn,n(𝜑2).
4.2 MTT matrix form
As already mentioned in the introduction, the Matrix Transfer Technique (MTT) is a
strategy for discretizing the fractional Laplace operator. Originally introduced in [21,
22] for the 1D case, the MTT has been also generalized to the two-dimensional case in
(17) Tn(f𝛼) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝓁0 𝓁−1 … … 𝓁1−n 𝓁1 𝓁0 𝓁−1 ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ ⋮ 𝓁1 𝓁0 𝓁−1 𝓁n−1 … … 𝓁1 𝓁0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , Tn,n(𝜑2) = tridiag(−In, Tn(f2) + 2In,−In) ∈ ℝ n2×n2 , 𝜑2(𝜃1, 𝜃2) = 4 − 2 cos 𝜃1− 2 cos 𝜃2. 𝜑𝛼(𝜃1, 𝜃2) = ( 4− 2 cos 𝜃1− 2 cos 𝜃2)𝛼∕2. (18) 𝜑𝛼(𝜃1, 𝜃2) = ( f2(𝜃1) + f2(𝜃2))𝛼∕2. (19) 𝜓𝛼(𝜃1, 𝜃2) =(f2(𝜃1))𝛼∕2+(f2(𝜃2))𝛼∕2; (20) Tn,n(𝜓𝛼) = Tn(f𝛼(𝜃1)) ⊗ In+ In⊗Tn(f𝛼(𝜃2)),
[44]. The approximation provided by this method is given by a fractional power of any reasonable discretization of the standard Laplacian. Here we consider two particular discretizations in the one-dimensional case: the one obtained by using centered finite differences and the one corresponding to a generic finite element approximation. We
denote by LFD
𝛼,n and L
FE
𝛼,n, the corresponding discrete fractional operators provided by
the MTT. In formulas (see Sect. 4.1)
where Mn and Kn are the usual finite element mass and stiffness matrices. In
particu-lar, Mn= hTn((2 + cos 𝜃)∕3) and Kn= h−1Tn(f2) when piecewise linear basis
func-tions are adopted.
Concerning the 2D case, in this work we will focus our spectral analysis on the fol-lowing MTT matrix forms:
where Mn,n and Kn,n denote the mass and stiffness matrices in the 2D case.
4.3 Matrix form for the case of non‑constant coefficient
Once we have a matrix representation for the 1D fractional Laplacian, say L𝛼,n ,
obtained as explained in the last two subsections (see (17), (21) and (22)), we can easily recover a discretization for the operator (13) on the uniform mesh (14) just taking
where Dn= diagj=1,…,n(D(𝜉j)) . Similar arguments apply to the d-dimensional
operator
In particular, a discretization of this operator when d = 2 takes the following form
where Dn,n is the diagonal matrix whose non-zero entries are the values of D(x) at
the grid points and L𝛼,n,n denotes a matrix form of the 2D fractional Laplacian.
(21) LFD 𝛼,n = −h −𝛼(T n(f2))𝛼∕2, (22) LFE 𝛼,n = −(M −1 n Kn) 𝛼∕2, LFD 𝛼,n,n= −h −𝛼(T n(f2) ⊗ I + I ⊗ Tn(f2) )𝛼∕2 = −h−𝛼(Tn,n(𝜑2)) 𝛼∕2, LFE 𝛼,n,n= −(M −1 n,nKn,n)𝛼∕2, (23) M 𝛼,n= DnL𝛼,n, −D(x)(−𝛥)𝛼∕2. M 𝛼,n,n= Dn,nL𝛼,n,n,
5 Spectral analysis
In this section, divided into three parts, the spectral analysis is performed following the
results reported in Sect. 2 concerning the Toeplitz technology and the quite powerful
GLT theory. The interesting message we can give is the following: independently of the used method, despite the substantial difference in the structure of the involved matrices,
the smallest eigenvalue scales like n−𝛼 and all the related matrix-sequences present the
same global eigenvalue distribution.
5.1 Toeplitz case
First we focus our attention on the spectral properties of Tn(f𝛼). Since its generating
function f𝛼 is an even, real-valued and nonnegative function, the matrix Tn(f𝛼) is
real, symmetric and positive semidefinite. Consequently,
where 𝛬n,𝛼 = diag(𝜆1,𝛼, 𝜆2,𝛼,… , 𝜆n,𝛼) is the diagonal matrix containing the
eigen-values of Tn(f𝛼) sorted in nondecreasing order and Qn,𝛼 is the orthogonal matrix
whose columns are the eigenvectors associated to the eigenvalues 𝜆j,𝛼, j= 1, 2, … , n.
In addition, considering that ess inf f𝛼= 0, ess sup f𝛼>0 and f𝛼∼|𝜃|𝛼, from
Theo-rem 3 we deduce that
• Tn(f𝛼) is positive definite for each n ∈ ℕ;
• the minimum eigenvalue of Tn(f𝛼) behaves as follows: 𝜆1,𝛼 ∼ 1∕n𝛼.
Now, using (20) and (24) we obtain that
The matrix (𝛬n,𝛼⊗In+ In⊗ 𝛬n,𝛼) is block diagonal with diagonal blocks, say Dj,
having the following form:
In particular, this implies that the minimum and the maximum eigenvalues of
Tn,n(𝜓𝛼) are, respectively,
It is worth to observe that since 𝜆1,𝛼 and 𝜆n,𝛼 are the minimum and the maximum
eigenvalues of Tn(f𝛼), the condition number in Euclidean norm of the matrix Tn,n(𝜓𝛼)
coincides with the one of the matrix Tn(f𝛼), i.e.,
In addition, the behavior of 𝜆1,𝛼 leads to deduce that
(24) Tn(f𝛼) = Qn,𝛼𝛬n,𝛼QTn,𝛼, Tn,n(𝜓𝛼) = (Qn,𝛼𝛬n,𝛼Q T n,𝛼) ⊗ In+ In⊗(Qn,𝛼𝛬n,𝛼Q T n,𝛼) = (Qn,𝛼⊗Qn,𝛼)(𝛬n,𝛼⊗In+ In⊗ 𝛬n,𝛼)(Qn,𝛼⊗Qn,𝛼) T . Dj= diag (𝜆1,𝛼+ 𝜆j,𝛼, 𝜆2,𝛼+ 𝜆j,𝛼,… , 𝜆n,𝛼+ 𝜆j,𝛼), j= 1, 2, … , n. 𝜆min(Tn,n(𝜓𝛼)) = 2 𝜆1,𝛼, 𝜆max(Tn,n(𝜓𝛼)) = 2 𝜆n,𝛼. 𝜇2(Tn,n(𝜓𝛼)) = 𝜇2(Tn(f𝛼)).
By virtue of the fact that f2∼|𝜃|2 (see (16)), the two generating functions defined in
(18) and (19) satisfy, respectively,
Using the polar coordinates (𝜌, 𝜂) we can write Therefore, from the previous relations we have
Since (|cos 𝜂|𝛼+|sin 𝜂|𝛼) is never identically zero, we conclude that
By applying the block version of Theorem 4 (see Theorem 3.9 in [40] for a more
general result), we have Consequently,
Notice that because of (25) and (26), both 𝜑𝛼 and 𝜓𝛼 have a zero at the origin of
order 𝛼 in the sense of Definition 2.
GLT considerations Thanks to Theorem 5 and item GLT3, we can immediately
conclude that {Tn(f𝛼)}n∼GLT,𝜆f𝛼, {Tn,n(𝜑𝛼)}n∼GLT,𝜆𝜑𝛼, {Tn,n(𝜓𝛼)}n∼GLT,𝜆𝜓𝛼. 5.2 MTT case
Finite Differences It is well-known that the discretization Tn(f2) given in (15) can
be diagonalized by means of the discrete sine transform. In formulas, Tn(f2) =
Qn,2𝛬n,2Qn,2, where the entries of the symmetric orthogonal matrix Qn,2 are given by
and, for j = 1, … , n, the non-zero entries of the diagonal matrix 𝛬n,2 are the
eignevalues 𝜆j,2= 2 − 2 cos(j𝜋∕(n + 1)). This enables to write directly the spectral
decomposition of LFD
𝛼,n by means of Qn,2𝛬
𝛼∕2
n,2Qn,2 and to conclude that
𝜆min(Tn,n(𝜓𝛼)) ∼ 1∕n 𝛼 . (25) 𝜑𝛼(𝜃1, 𝜃2) ∼(|𝜃1|2+|𝜃 2|2 )𝛼∕2 , 𝜓𝛼(𝜃1, 𝜃2) ∼|𝜃1|𝛼+|𝜃 2| 𝛼 . |𝜃1| = 𝜌|cos 𝜂|, |𝜃2| = 𝜌|sin 𝜂|. ( |𝜃1|2+|𝜃2|2 )𝛼∕2 = 𝜌𝛼, |𝜃1|𝛼+|𝜃2|𝛼= 𝜌𝛼(|cos 𝜂|𝛼+|sin 𝜂|𝛼). (26) ( |𝜃1|2+|𝜃2|2 )𝛼∕2 ∼|𝜃1|𝛼+|𝜃2|𝛼. 𝜆min(Tn,n(𝜑𝛼)) ∼ 𝜆min(Tn,n(𝜓𝛼)). 𝜆min(Tn,n(𝜑𝛼)) ∼ 1∕n𝛼. (Qn,2)ij= √ 2 n+ 1sin ( ij𝜋 n+ 1 ) , i, j= 1, … , n, 𝜆min(−h 𝛼LFD 𝛼,n ) = (𝜆1,2) 𝛼∕2∼ 1 n𝛼.
Similarly, in two dimensions the matrix Tn,n(𝜑2) = Tn(f2) ⊗ In+ In⊗Tn(f2)
can be diagonalized by means of Qn,2⊗Qn,2, and its diagonal form is
In⊗ 𝛬n,2+ 𝛬n,2⊗In . Therefore, its eigenvalues are all the possible sums 𝜆i,2+ 𝜆j,2 ,
with i, j = 1, … , n and we can compute LFD
𝛼,n,n by means of the decomposition
(Qn,2⊗Qn,2)(In⊗ 𝛬n,2+ 𝛬n,2⊗In)𝛼∕2(Qn,2⊗Qn,2) which implies that
and 𝜇2(L
FD
𝛼,n) = 𝜇2(L
FD 𝛼,n,n).
GLT considerations Since Tn(f2) is a symmetric positive definite matrix and
{Tn(f2)}n∼GLTf2 by GLT3, we can use GLT6 to conclude that {−h𝛼LFD
𝛼,n}n∼GLTf𝛼 .
Moreover, LFD
𝛼,n is still real symmetric and by GLT1 we also have {−h
𝛼LFD
𝛼,n}n ∼𝜆f𝛼 .
In other words, {−h𝛼LFD
𝛼,n}n is a GLT matrix-sequence that shares the same symbol
of the Toeplitz sequence {Tn(f𝛼)}n obtained by a direct centered finite difference
dis-cretization of the fractional Laplacian. As a consequence, by GLT2
which thanks to GLT5 and Proposition 1 is equivalent to say that the
matrix-sequence {−h𝛼LFD
𝛼,n − Tn(f𝛼)}n is weakly clustered at zero.
A similar reasoning proves that {−h𝛼LFD
𝛼,n,n}n∼GLT,𝜆𝜑𝛼 and that
{−h𝛼LFD
𝛼,n,n− Tn,n(𝜑𝛼)}n∼GLT,𝜎0.
Finite Elements The matrix representation of the 1D standard Laplacian resulting
from the finite element method is given by Cn= M−1n Kn. Both Mn and Kn are real
and symmetric and Mn is positive definite. Considering that ̃Cn= M
−1∕2
n KnM
−1∕2
n is a
real symmetric matrix and
it follows that Cn is diagonalizable. In particular, when piecewise linear basis
func-tions are adopted, that is Mn= hTn((2 + cos 𝜃)∕3) and Kn= h−1Tn(f2), then Cn can
be still diagonalized by means of Qn,2 and the eigenvalues of both Cn and L
FE 𝛼,n are
known exactly. Specifically, we have
Similar considerations can be made in the two-dimensional setting. The general case
of high-order finite elements is less trivial. For example, in [14] it has been shown
that the matrices Mn and Kn do not belong to the same matrix algebra when
consid-ering finite elements with maximum regularity of order greater than or equal to 3. In
this context, an approximation of the eigenvalues of LFE
𝛼,n and L
FE
𝛼,n,n can be obtained
using the notion of symbol, as outlined in the next paragraph.
GLT considerations Here we discuss the case of high-order finite elements with
maximum regularity and we exploit the results in [15] where it has been shown that
{h−1M
n}n and {hKn}n are both GLT matrix-sequences. We denote the corresponding
𝜆min(−h 𝛼LFD 𝛼,n,n) = (2𝜆1,2) 𝛼∕2∼ 1 n𝛼, {−h𝛼LFD 𝛼,n − Tn(f𝛼)}n∼GLT0, (27) Cn= Mn−1∕2C̃nMn1∕2, 𝜆min(−h𝛼LFE 𝛼,n) = ( 𝜆 min(hKn) 𝜆max(h−1M n) )𝛼∕2 = ( 2− 2 cos( 𝜋 n+1) 2 3+ 1 3cos( n𝜋 n+1) )𝛼∕2 ∼ 1 n𝛼.
symbols as m and 𝜅, respectively. Since both Mn and Kn are real symmetric, by GLT1
the matrix-sequences {h−1M
n}n and {hKn}n distribute as m, 𝜅 also in the eigenvalue
sense. In formulas, Since h−1M
n is positive definite with minimal eigenvalue well separated from
zero (see again [15]), the corresponding function m is invertible everywhere. For
instance, when using piecewise linear elements, h−1M
n= Tn((2 + cos 𝜃)∕3) and
hence
on the whole definition domain. Consequently, item GLT2 implies that {h2C
n}n∼GLTm−1𝜅=∶ 𝜒. In addition, from (27) and by recalling that ̃Cn is real
symmetric, by using items GLT1, GLT2, and GLT6 we get
Furthermore, since Kn is positive semidefinite, also ̃Cn is positive semidefinite and
then ( ̃Cn)𝛼∕2 is well-defined. Taking into account that ( ̃Cn)𝛼∕2 is still real symmetric,
by GLT6 we infer that
To prove that also {−h𝛼LFE
𝛼,n}n is a GLT matrix-sequence we first observe that (see
(22) and (27))
The last equality in (30) comes directly from the definition of matrix function and
from the fact that ̃Cn is symmetric positive semidefinite. More precisely, if we write
the diagonal form of ̃Cn as ̃Cn= ZJZT , then
Starting from (30) and putting together (28), (29), and by using items GLT1, GLT2,
and GLT6, we deduce that
Notice that the above reasoning still works in the two-dimensional setting. When replacing Mn and Kn with Mn,n and Kn,n we have
(28) {h−1Mn}n∼GLT,𝜆m, {hKn}n∼GLT,𝜆𝜅. m= (2 + cos 𝜃)∕3≥ 1∕3 > 0 {h2C̃ n}n∼GLT,𝜆𝜒. (29) {h𝛼( ̃C n)𝛼∕2}n∼GLT,𝜆𝜒𝛼∕2. (30) −L𝛼FE,n= (Mn−1∕2C̃nMn1∕2)𝛼∕2= Mn−1∕2( ̃Cn)𝛼∕2M1∕2n . (Mn−1∕2C̃nMn1∕2)𝛼∕2= (M−1∕2 n ZJZ TM1∕2 n ) 𝛼∕2 = (M−1∕2n ZJ(M−1∕2n Z)−1)𝛼∕2 = M−1∕2n Z(J)𝛼∕2ZTM1∕2 n = M−1∕2n ( ̃Cn)𝛼∕2Mn1∕2. (31) {−h𝛼LFE 𝛼,n}n∼GLT,𝜆𝜒 𝛼∕2.
with 𝜁 = m−1𝜅 where now m, 𝜅 are the symbols of the properly scaled
two-dimen-sional mass and stiffness matrices.
As an example, we consider the finite element method constructed by using C2
piecewise cubic basis functions. Since
and
in this case while
Remark 2 In the case of high-order finite element with a certain fixed regularity the resulting matrix-sequences are (sequences of submatrices of) block GLTs. Analo-gous results to (31) and (32) follow using the multilevel block GLT theory [5, 6, 16].
5.3 Non‑constant coefficient case
In this section we review how the GLT theory can be used for studying the spectral
behavior of the matrix-sequence {−h𝛼M
𝛼,n}n, with M𝛼,n given by (23). In
particu-lar, we focus on the case of the finite differences which has already been analyzed
in [13, Proposition 5] and in [29, Proposition 3.11] and on the finite element one.
Everything works in the two-dimensional setting as well, so we skip the details for this case.
Let h𝛼(x, 𝜃) = D(x)g𝛼(𝜃) be the symbol associated to −h𝛼M𝛼,n= −h𝛼DnL𝛼,n,
where L𝛼,n is one of the discretizations of the 1D fractional Laplacian discussed in
the previous sections whose symbol is denoted by g𝛼 . Noticing that Dn is a positive
definite diagonal matrix, then
that is M𝛼,n is similar to a real symmetric matrix. Therefore, all the eigenvalues of
M𝛼
,n are real and by GLT1 we infer
{
−h𝛼M
𝛼,n
}
n∼𝜆( ̂h𝛼(̂x, 𝜃), [0, 1] × [−𝜋, 𝜋]),
where ̂h𝛼(̂x, 𝜃) = h𝛼(a + (b − a)̂x, 𝜃). After the affine change of variables
(32) {−h𝛼LFE 𝛼,n,n}n∼GLT,𝜆 𝜁𝛼∕2, m(𝜃) = 151 315+ 397 840cos 𝜃+ 1 21cos 2𝜃+ 1 2520cos 3𝜃 𝜅(𝜃) = 2 3− 1 4cos 𝜃− 2 5cos 2𝜃− 1 60cos 3𝜃, {−h𝛼LFE 𝛼,n}n∼GLT,𝜆(m−1𝜅)𝛼∕2 {−h𝛼LFE 𝛼,n,n}n ∼GLT,𝜆 (𝜅 (𝜃1)m(𝜃2) + m(𝜃1)𝜅(𝜃2) m(𝜃1)⋅m(𝜃2) )𝛼∕2 . D−1∕2 n M𝛼,nD 1∕2 n = D 1∕2 n L𝛼,nD 1∕2 n ,
x= a + (b − a)̂x,{−h𝛼M 𝛼,n } n∼𝜆 ̂h𝛼(̂x, 𝜃) is equivalent to { −h𝛼M 𝛼,n } n∼𝜆h𝛼(x, 𝜃). Therefore,
From the limit relation
which is true simply because
it is clear that the symbol h𝛼 has a zero of order 𝛼 at 𝜃 = 0 , whenever the diffusion
coefficient D is bounded and positive.
Summary of spectral results We now summarize all the spectral results obtained
by collecting them in two groups, those relating to the one-dimensional case and those relating to the two-dimensional case. Furthermore, we assign to each operator an acronymous that will be useful in the next section.
• 1D case: 1FD-T: {Tn(f𝛼)}n ∼GLT,𝜆f𝛼, f𝛼= (2 − 2 cos 𝜃)𝛼∕2 1FD-MTT: {−h𝛼LFD 𝛼,n}n∼𝜆f𝛼 1FE-MTT: {−h𝛼LFE 𝛼,n}n∼𝜆(m−1𝜅)𝛼∕2=∶ 𝜒𝛼∕2
• piecewise linear basis functions:
• C2 piecewise cubic basis functions:
1FD-NC: {−h𝛼M 𝛼,n}n∼GLT,𝜆h𝛼(x, 𝜃) = D(x)g𝛼(𝜃) with g𝛼= f𝛼, 𝜒𝛼∕2 • 2D case: 2FD-T: {Tn,n(𝜑𝛼)}n∼GLT,𝜆 𝜑𝛼, 𝜑𝛼= (2 − 2 cos 𝜃1− 2 cos 𝜃2)𝛼∕2 2FD-T-S: {Tn,n(𝜓𝛼)}n∼GLT,𝜆𝜓𝛼, 𝜓𝛼 = (2 − 2 cos 𝜃1)𝛼∕2+ (2 − 2 cos 𝜃2)𝛼∕2 2FD-MTT: {−h𝛼LFD 𝛼,n,n}n∼𝜆𝜑𝛼 2FE-MTT: {−h𝛼LFE 𝛼,n,n}n ∼𝜆((m(𝜃1, 𝜃2))−1𝜅(𝜃1, 𝜃2))𝛼∕2=∶ 𝜁𝛼∕2 { −h𝛼M𝛼 ,n } n∼GLT,𝜆h𝛼(x, 𝜃). lim 𝜃→0 h𝛼(x, 𝜃) g𝛼(𝜃) = D(x), h𝛼(x, 𝜃) g𝛼(𝜃) = D(x), ∀(x, 𝜃), m(𝜃) = 2 3+ 1 3cos 𝜃, 𝜅(𝜃) = 2 − 2 cos 𝜃 m(𝜃) = 151 315+ 397 840cos 𝜃+ 1 21cos 2𝜃+ 1 2520cos 3𝜃 𝜅(𝜃) = 2 3− 1 4cos 𝜃− 2 5cos 2𝜃− 1 60cos 3𝜃
6 Numerical results
In this section we numerically check some of the relations we have listed at the end of the previous section. In all our experiments we used the Matlab function eig to compute the eigenvalues. In addition, when finite elements are involved, we opt for
C2 piecewise cubic basis functions.
We start by verifying the spectral relations 1FD-T and 1FE-MTT. Since the sym-bols involved are even functions, we can restrict to the interval [0, 𝜋] and consider a
uniform mesh with step-size h = 𝜋∕(n + 1). As shown in Fig. 1, setting 𝛼 = 1.3 and
n= 100, there is a good matching between the eigenvalues of the matrices Tn(f𝛼) and
−h𝛼LFE
𝛼,n , and a sampling of f𝛼 on the points grid.
Similar results can be obtained when checking 1FD-NC with g𝛼= f𝛼 , 𝛼 = 1.2,
n= 100, and D(x) = 𝛤 (3 − 𝛼)x𝛼 or D(x) = exp(8x) (refer to Fig. 2).
Consider now the square domain [0, 𝜋]2 and define a uniform mesh with step-size
h= 𝜋∕(n + 1) in each domain direction. In Fig. 3 we check the validity of 2FD-T-S
and 2FE-MTT comparing the spectrum of the matrices Tn,n(𝜓𝛼) , −h𝛼L
FE
𝛼,n,n with a
sampling of 𝜓𝛼 , 𝜁𝛼∕2 , when 𝛼 = 1.7 and n = 10. Also in this two-dimensional
exam-ple the symbol is describing quite well the spectrum of the considered discretization matrices.
As a final test, in Fig. 4 we compare the contour plots of all the symbols
cor-responding to two-dimensional discretization matrices, i.e., 𝜓𝛼 , 𝜁𝛼∕2 , and 𝜑𝛼 , when
𝛼= 1.4 . What can be immediately noticed is that the symbol 𝜓𝛼 shows sharper
pat-terns than 𝜑𝛼 and especially 𝜁𝛼∕2 , when approaching zero. Moreover, we note that
the 2FE-MTT symbol is close to zero on a slightly larger portion of the domain than the other two symbols. This results in a worse ill-conditioning of the cor-responding discretization matrices and refers to the degree of the approximation space. 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 eigs symbol 0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 eigs symbol
Fig. 1 1D case—comparison between the spectrum of the matrices Tn(f𝛼), −h𝛼L FE
𝛼,n and a sampling of the related symbols f𝛼 and 𝜒𝛼∕2 for n = 100 and 𝛼 = 1.3
0 20 40 60 80 100 0 0.5 1 1.5 2 2.5 eigs symbol 0 20 40 60 80 100 0 1000 2000 3000 4000 5000 6000 7000 eigs symbol (a) (b)
Fig. 2 1D non-constant coefficient case—comparison between the spectrum of the matrix DnTn(f𝛼) and a sampling of the related symbol D(x)f𝛼(𝜃) for n = 100 and 𝛼 = 1.2
0 20 40 60 80 100 0 1 2 3 4 5 6 7 eigs symbol 0 20 40 60 80 100 0 2 4 6 8 10 12 eigs symbol
Fig. 3 2D case—comparison between the spectrum of the matrices Tn,n(𝜓𝛼) , −h 𝛼LFE
𝛼,n,n and a sampling of the related symbols 𝜓𝛼 and 𝜁𝛼∕2 for n = 10 and 𝛼 = 1.7
-2 0 2 -3 -2 -1 0 1 2 3 1 2 3 4 -2 0 2 -3 -2 -1 0 1 2 3 1 2 3 4 5 6 7 -2 0 2 -3 -2 -1 0 1 2 3 0.5 1 1.5 2 2.5 3 3.5 (a) (b) (c)
7 Conclusions
Starting from two different ways of defining the fractional Laplace operator in more than one dimension we have studied the spectral properties of related finite differ-ence and finite element discretization matrices. Precisely, we have analyzed the spectrum/conditioning of
• the Toeplitz matrices whose symbol is a proper fraction of the symbol of the
standard Laplacian operator;
• the matrices obtained by the MTT as fractional powers of both finite difference
and finite element approximations of the standard Laplacian.
Furthermore, we have shown that the corresponding matrix-sequences belong to the GLT class. The characterization of all the aforementioned discretization matrix-sequences as GLT matrix-matrix-sequences relates to the solution of the corresponding
linear systems. Inspired by the GLT-based algorithmic proposals in [29] for finite
difference discretizations of the 2D fractional Laplacian and due to the ill-condition-ing observed when considerill-condition-ing the 2FE-MTT symbol, in a future work it could be interesting to exploit the GLT structure for the definition of preconditioners suitable for finite element discretizations.
Acknowledgements Open access funding provided by Università di Pisa within the CRUI-CARE Agreement.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.
References
1. Aceto, L., Bertaccini, D., Durastante, F., Novati, P.: Rational Krylov methods for functions of matri-ces with applications to fractional partial differential equations. J. Comput. Phys. 396, 470–482 (2019)
2. Aceto, L., Novati, P.: Efficient implementation of rational approximations to fractional differential operators. J. Sci. Comput. 76(1), 651–671 (2018)
3. Aceto, L., Novati, P.: Rational approximation to fractional powers of self-adjoint positive operators. Numer. Math. 143(1), 1–16 (2019)
4. Aceto, L., Novati, P.: Rational approximation to the fractional Laplacian operator in reaction-diffu-sion problems. SIAM J. Sci. Comput. 39, A214–A228 (2017)
5. Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block Generalized Locally Toeplitz Sequences: Theory and Applications in the Unidimensional Case, Technical Report, N. 4, July 2019, Depart-ment of Information Technology, Uppsala University. http://www.it.uu.se/resea rch/publi catio ns/ repor ts/2019-004/
6. Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block Generalized Locally Toeplitz Sequences: Theory and Applications in the Multiidimensional Case, Technical Report, N. 5, July 2019, Depart-ment of Information Technology, Uppsala University. http://www.it.uu.se/resea rch/publi catio ns/ repor ts/2019-005/
7. Bhatia, R.: Matrix Analysis. Springer, New York (1997)
8. Bonito, A., Pasciak, J.E.: Numerical approximation of fractional powers of elliptic operators. Math. Comput. 84(295), 2083–2110 (2015)
9. Böttcher, A., Grudsky, S.: On the condition numbers of large semi-definite Toeplitz matrices. Linear Algebra Appl. 279(1/3), 285–301 (1998)
10. Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)
11. Çelik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)
12. Defterli, O., D’Elia, M., Du, Q., Gunzburger, M., Lehoucq, R., Meerschaert, M.M.: Fractional diffu-sion on bounded domains. Fract. Calc. Appl. Anal. 18(2), 342–360 (2015)
13. Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving precondi-tioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)
14. Ekström, S.E., Furci, I., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Are the eigenval-ues of the B-spline isogeometric analysis approximation of −Δu = 𝜆u known in almost closed form? Numer. Linear Algebra Appl. 25(5), e2198 (2018)
15. Garoni, C., Manni, C., Pelosi, F., Serra-Capizzano, S., Speleers, H.: On the spectrum of stiffness matrices arising from isogeometric analysis. Numer. Math. 127, 751–799 (2014)
16. Garoni, C., Mazza, M., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: from the theory to the applications. Axioms 7(3), 49 (2018)
17. Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. I. Springer, Cham (2017)
18. Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. II. Springer, Cham (2018)
19. Grenander, U., Szegö, G.: Toeplitz Forms and their Applications, 2nd edn. Chelsea, New York (1984)
20. Harizanov, S., Lazarov, R., Marinov, P., Margenov, S., Pasciak, J.: Comparison analysis on two numerical methods for fractional diffusion problems based on rational approximations of
t𝛾, 0≤ t ≤ 1,arXiv :1805.00711 v1
21. Ilić, M., Liu, F., Turner, I., Anh, V.: Numerical approximation of a fractional-in-space diffusion equation I. Frac. Calc. App. Anal. 8, 323–341 (2005)
22. Ilić, M., Liu, F., Turner, I., Anh, V.: Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal. 9, 333–349 (2006)
23. Jin, X.-Q., Lin, F.-R., Zhao, Z.: Preconditioned iterative methods for two-dimensional space-frac-tional diffusion equations. Commun. Comput. Phys. 18(2), 469–488 (2015)
24. Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20, 7–51 (2017)
25. Lei, S.-L., Sun, H.-W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)
26. Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015)
27. Lubich, C.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45, 463–469 (1985)
28. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)
29. Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.: Spectral analysis and multigrid precondi-tioners for two-dimensional space-fractional diffusion equations. J. Comput. Phys. 350, 992–1011 (2017)
30. Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. (2006)
31. Ortigueira, M.D.: Fractional central differences and derivatives. J. Vib. Control 14, 1255–1266 (2008)
32. Pang, H.-K., Sun, H.-W.: Multigrid method for fractional diffusion equations. J. Comput. Phys.
231(2), 693–703 (2012)
33. Podlubny, I.: Fractional Differential Equations: Mathematics in Science and Engineering, vol. 198. Academic Press Inc, San Diego, CA (1999)
34. Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3, 359–386 (2000)
35. Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y.Q., Vinagre Jara, B.M.: Partial fractional dif-ferential equations: Matrix approach to discrete fractional calculus II. J. Comput. Phys. 228, 3137– 3153 (2009)
36. Popolizio, M.: A matrix approach for partial differential equations with Riesz space fractional deriv-atives. Eur. Phys. J. Spec. Top. 222, 1975–1985 (2013)
37. Riesz, M.: Intégrales de Riemann-Liouville et potentiels. Acta Sci. Math. Szeged 9, 1–42 (1938) 38. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivatives-Theory and
Applica-tions. Gordon and Breach Science Publishers, New York (1987)
39. Serra, S.: On the extreme eigenvalues of Hermitian (block) Toeplitz matrices. Linear Algebra Appl.
270, 109–129 (1998)
40. Serra, S.: On the extreme eigenvalues of Hermitian (block) Toeplitz matrices. SIAM J. Matrix Anal. Appl. 20(1), 31–44 (1998)
41. Tilli, P.: A note on the spectral distribution of Toeplitz matrices. Linear Multilin. Algebra 45(2/3), 147–159 (1998)
42. Tyrtyshnikov, E.: A unifying approach to some old and new theorems on distribution and clustering. Linear Algebra Appl. 232, 1–43 (1996)
43. Zamarashkin, N.L., Tyrtyshnikov, E.: Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function. Sbornik: Math. 188(8), 1191 (1997)
44. Yang, Q., Turner, I., Liu, F., Ilić, M.: Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. SIAM J. Sci. Comput. 33(3), 1159–1180 (2011)
45. Yu, Q., Liu, F., Turner, I., Burrage, K.: Numerical investigation of three types of space and time fractional Bloch–Torrey equations in 2D. Cent. Eur. J. Phys. 11(6), 646–665 (2013)
46. Zoia, A., Rosso, A., Kardar, M.: Fractional Laplacian in bounded domains. Phys. Rev. E 76, 021116 (2007)
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published
